WAEC 2022 · Paper 2 · Q6

(a)
Given that $P = \{x : x \ge -2\}$ , $Q = \{x : 1 < x < 6\}$ and $R = \{x : x < 3\}$ , where $x$ is an integer, find: (i) $P \cap (Q \cup R)$ ; (ii) $(P \cap Q) \cup (P \cap R)$ .
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Both are $\{-2, -1, 0, 1, 2, 3, 4, 5\}$
(b)
A money lender lends at a rate of $15\%$ per annum simple interest. In how many years, correct to the nearest year, will the interest be the same as amount borrowed?

Worked solution (try it first)

(a)

List the integers: $P = \{-2, -1, 0, 1, \ldots\}$ , $Q = \{2, 3, 4, 5\}$ , $R = \{\ldots, 0, 1, 2\}$ .

(i)

$Q \cup R = \{\ldots, 0, 1, 2, 3, 4, 5\}$ , so $P \cap (Q \cup R) = \{-2, -1, 0, 1, 2, 3, 4, 5\}$ .

(ii)

$P \cap Q = \{2, 3, 4, 5\}$ and $P \cap R = \{-2, -1, 0, 1, 2\}$ , so $(P \cap Q) \cup (P \cap R) = \{-2, -1, 0, 1, 2, 3, 4, 5\}$ , the same set.

(b)

The interest equals the amount borrowed: $\frac{P \times 15 \times T}{100} = P$ , so $15T = 100$ and $T = 6\frac23 \approx 7$ years.

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